EMI P. No.

- 1

JRS TUTORIALS
PHYSICS PRACTICE SHEET
Electromagnetic Induction
1. The flux of magnetic field through a closed conducting loop of resistance 0.4  changes with
time according to the equation 0.20t 2 + 0.40t + 0.60 where t is time in seconds. Find
(i) the induced emf at t = 2s.
(ii) the average induced emf in t = 0 to t = 5 s.
(iii) charge passed through the loop in t = 0 to t = 5s
(iv) average current in time interval t = 0 to t = 5 s (v) heat produced in t = 0 to t = 5s.
2. Figure illustrates plane figures made of thin conductors which are located in a uniform magnetic
field directed away from a reader beyond the plane of the drawing. The magnetic induction starts
diminishing. Find how the currents induced in these loops are directed.

3. Two straight long parallel conductors are moved towards each other. A constant current i is
flowing through one of them. What is the direction of the current induced in other conductor?
What is the direction of induced current when the conductors are drawn apart.
4. A metallic ring of area 25 cm2 is placed perpendicular to a magnetic field of 0.2 T .It is
removed from the field in 0.2 s. Find the average emf produced in the ring during this time .
5. A conducting circular loop of radius R confined in a plane is rotated in its own plane with
some angular velocity . A uniform magnetic field B exist in the region. Find the current
induced in the loop
6. A heart pacing device consists of a coil of 50 turns & radius 1 mm just inside the body with
a coil of 1000 turns & radius 2 cm placed concentrically and coaxially just outside the body.
Calculate the average induced EMF in the internal coil, if a current of 1A in the external coil
collapses in 10 milliseconds.
7. A solenoid has a cross sectional area of 6.0×10–4 m2, consists of 400 turns per meter, and carries
a current of 0.40 A. A 10 turn coil is wrapped tightly around the circumference of the solenoid. The
ends of the coil are connected to a 1.5  resistor. Suddenly, a switch is opened, and the current in
the solenoid dies to zero in a time 0.050 s. Find the average current in the coil.
8. A uniform magnetic field of 0.08 T is directed into the plane of the page and perpendicular to it as
shown in the figure. A wire loop in the plane of the page has constant area 0.010 m2. The magnitude
of magnetic field decrease at a constant rate of 3.0 × 10–4 Ts–1. Find the magnitude and direction
of the induced emf in the loop.

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EMI P. No.- 2
9. A conducting circular loop is placed in a uniform magnetic field of 0.02 T, with its plane perpendicular
to the field . If the radius of the loop starts shrinking at a constant rate of 1.0 mm/s, then find the emf
induced in the loop, at the instant when the radius is 4 cm.
10. Figure shows a square loop of resistance 1  of side 1 m being moved towards right at a
constant speed of 1 m/s. The front edge enters the 3 m wide magnetic field ( B = 1 T ) at t =
0. Draw the graph of current induced in the loop as time passes. (Take anticlockwise direction
of current as positive). Find the total heat produced in the loop of the during the interval 0 to 5 s

11. Two concentric and coplanar circular coils have radii a and b(>>a)as shown in figure. Resistance
of the inner coil is R. Current in the outer coil is increased from 0 to i , then find the total charge
circulating the inner coil.

12. A closed coil having 50 turns is rotated in a uniform magnetic field B = 2 ×10 – 4 T about a
diameter which is perpendicular to the field. The angular velocity of rotation is 300 revolutions
per minute. The area of the coil is 100 cm 2 and its resistance is 4  Find
(a) the average emf developed in half a turn from a position where the coil is perpendicular to
the magnetic field,
(b) the average emf in a full turn ,
(c) the net charge flown in part (a) and
(d) the emf induced as a function of time if it is zero at t=0 and is increasing in positive
direction.
(e) the maximum emf induced.
(f) the average of the squares of emf induced over a long period
13. A -shaped conductor is located in a uniform magnetic field perpendicular to the plane of the
dB
conductor and varying with time at the rate = 0.10 T/s. A conducting connector starts
dt
moving with a constant acceleration w = 10 cm/s2 along the parallel bars of the conductor.
The length of the connector is equal to  = 20 cm. Find the emf induced in the loop t = 2.0 s
after the beginning of the motion, if at the moment t = 0 the loop area and the magnetic
induction are equal to zero. The self inductance of the loop is to be neglected.
)
XB

)
14. A uniform but time varying magnetic field B = Kt – C ; (0  t  C/K), where K and C are constants
and t is time, is applied perpendicular to the plane of the circular loop of radius ‘a’ and resistance
R. Find the total charge that will pass around the loop.
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EMI P. No.- 3
15. Two infinite long straight parallel wires A and B are separated by 0.1 m distance and carry
equal current in opposite directions. A square loop of wire C of side 0.1 m lies in the plane A
and B. The loop of wire C is kept parallel to both A and B at a distance of 0.1 m from the
nearest wire. Calculate the EMF induced in loop C while the current in A and B is increasing
at the same rate of 103 As1. Also indicate the direction of current in loop C

16. A plane spiral with a great number N of turns wound tightly to one another is located in a
uniform magnetic field perpendicular to the spiral’s plane. The outside radius of the spiral’s
turns is equal to a and inner radius is zero. The magnetic induction varies with time as B = B0
sin t, where B0 and  are constants. Find the amplitude of emf induced in the spiral.
17. A long straight wire is arranged along the symmetry axis of a toroidal coil of rectangular
crosssection, whose dimensions are given in the figure. The number of turns on the coil is N, and
relative permeability of the surrounding medium is unity. Find the amplitude of the emf induced in
this coil, if the current i = im cos t flows along the straight wire.

18. Figure shows a square frame of wire having a total resistance r placed coplanarly with a long,
straight wire. The wire carries a current i given by i = i0 cos (2t/T). Find (a) the flux of the magnetic
field through the square frame, (b) the emf induced in the frame and (c) the heat developed in the
frame in the time interval 0 to 10 T.
a

b
19. A wire loop enclosing a semi-circle of radius a is located on the boundary of a uniform magnetic
field of induction B (Figure). At the moment t = 0 the loop is set into rotation with a constant
angular acceleration  about an axis O coinciding with a line of vector B on the boundary.
Find the emf induced in the loop as a function of time t. Draw the approximate plot of this
function. The arrow in the figure shows the emf direction taken to be positive. (at t = 0 loop
was completely outside)

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EMI P. No.- 4
20. In the figure, a long thin wire carrying a varying current i = i0 sin t lies at a distance y above one
edge of a rectangular wire loop of length L and width W lying in the x-z plane. What emf is induced
in the loop.

21. A rectangular frame ABCD made of a uniform metal wire has a straight connection between E &
F made of the same wire as shown in the figure. AEFD is a square of side 1 m & EB = FC = 0.5
m. The entire circuit is placed in a steadily increasing uniform magnetic field directed into the place
of the paper & normal to it . The rate of change of the magnetic field is 1 T/s, the resistance per
unit length of the wire is 1 /m. Find the current in segments AE, BE & EF.

22. A wire is bent into 3 circular segments of radius r = 10 cm as shown in figure . Each segment is a
quadrant of a circle, ab lying in the xy plane, bc lying in the yz plane & ca lying in the zx plane.

(i) if a magnetic field B points in the positive x direction, what is the magnitude of the emf developed
in the wire, when B increases at the rate of 3 mT/s ?
(ii) what is the direction of the current in the segment bc.
23. A right angled triangle abc, made from a metallic wire, moves at a uniform speed v in its plane
as shown in the figure. A uniform magnetic field B exists in the perpendicular direction. Find
the emf induced (a) in the loop abc, (b) in the segment bc, (c) in the segment ac and (d) in the
segment ab.

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EMI P. No.- 5
24. A metallic metre stick translates in a direction making an angle of 60º with its length. The plane
of motion is perpendicular to a uniform magnetic field of 0.1 T that exists in the space. Find
the emf induced between the ends of the rod if the speed of translation is 0.2 m/s.
 B0
25. The magnetic field in a region is given by B = k̂ x where L is a fixed length. A conducting rod
L
of length L lies along the X-axis between the origin and the point (L, 0, 0). If the rod moves with a

velocity v  v 0 ĵ , find the emf induced between the ends of the rod.
26. A square frame of wire abcd of side 1 m has a total resistance of 4 . It is pulled out of a magnetic
field B = 1 T by applying a force of 1 N (figure). It is found that the frame moves with constant
speed. Find
(a) this constant speed,
(b) the emf induced in the loop,
(c) the potential difference between the points a and b and
(d) the potential difference between the points c and d.
27. A straight wire with a resistance of r per unit length is bent to form an angle 2. A rod of the same
wire perpendicular to the angle bisector (of 2) forms a closed triangular loop. This loop is placed
in a uniform magnetic field of induction B. Calculate the current in the loop when the rod moves at
a constant speed V.

28. A circular conduting-ring of radius r translates in its plane with a constant velocity v. A uniform
magnetic field B exists in the space in a direction perpendicular to the plane of the ring. Consider
different pairs of diametrically opposite points on the ring. (a) Between which pair of points is the
emf maximum? (b) Between which pair of points is the emf minimum? What is the value of this
minimum emf?
29. A wire forming one cycle of sine curve is moved in x-y plane with velocity
 
V  Vx i  Vy j . There exist a magnetic field B   B 0 k . Find the motional
emf develop across the ends PQ of wire.
30. A wire bent as a parabola y = kx2 is located in a uniform magnetic field of induction B, the
vector B being perpendicular to the plane x, y. At the moment t = 0 a connector starts sliding
translation wise from the parabola apex with a constant acceleration a (figure). Find the emf
of electromagnetic induction in the loop thus formed as a function of y.
y
B
a +

0 x
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EMI P. No.- 6
31. A rectangular loop with a sliding connector of length l = 1.0 m is situated in a uniform magnetic field
B = 2T perpendicular to the plane of loop. Resistance of connector is r = 2. Two resistances of
6 and 3 are connected as shown in figure. Find the external force required to keep the connector
moving with a constant velocity v = 2m/s.

32. The horizontal component of the earth’s magnetic field at a place is 3 × 10–4 T and the dip is tan–
1(4/3). A metal rod of length 0.25 m placed in the north-south position is moved at a constant

speed of 10 cm/s towards the east. Find the e.m.f. induced in the rod.
33. A rectangular loop of dimensions l & w and resistance R moves with constant velocity V to the
right as shown in the figure. It continues to move with same speed through a region containing a
uniform magnetic field B directed into the plane of the paper & extending a distance 3 W. Sketch
the flux, induced emf & external force acting on the loop as a function of the distance.

34. Two straight conducting rails form a right angle where their ends are joined. A conducting bar
contact with the rails starts at vertex at the time t = 0 & moves symmetrically with a constant
velocity of 5.2 m/s to the right as shown in figure. A 0.35 T magnetic field points out of the page.
Calculate:
(i) The flux through the triangle by the rails & bar at t = 3.0 s.
(ii) The emf around the triangle at that time.
(iii) In what manner does the emf around the triangle vary with time .
35. Consider the possibility of a new design for an electric train. The engine is driven by the force due
to the vertical component of the earths magnetic field on a conducting axle. Current is passed
down one coil, into a conducting wheel through the axle, through another conducting wheel & then
back to the source via the other rail.
(i) What current is needed to provide a modest 10  KN force ? Take the vertical component of the
earth's field be 10 T & the length of axle to be 3.0 m .
(ii) How much power would be lost for each  of resistivity in the rails ?
(iii) Is such a train realistic ?
36. A long straight wire carries a current 0. at distance a and b from it there are two other wires,
parallel to the former one, which are interconnected by a resistance R (figure). A connector
slides without friction along the wires with a constant velocity v. Assuming the resistances of
the wires, the connector, the sliding contacts, and the self-inductance of the frame to be
negligible, find

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EMI P. No.- 7
(a) The magnitude and the direction of the current induced in the connector;
(b) The force required to maintain the connector’s velocity constant.
(c) Point of application (distance from the long wire) of magnetic force on sliding wire due to
the long wire.

37. A metal rod of length 15 × 10–2 m rotates about an axis passing through one end with a
uniform angular velocity of 60 rad s–1. A uniform magnetic field of 0.1 Tesla exists in the
direction of the axis of rotation. Calculate the EMF induced between the ends of the rod.
38. A thin wire of negligible mass & a small spherical bob constitute a simple pendulum of effective
length . If this pendulum is made to swing through a semi-vertical angle , under gravity in a
plane normal to a uniform magnetic field of induction B, find the maximum potential difference
between the ends of the wire.
39. A metal rod of resistance 20 is fixed along a diameter of a conducting ring of radius 0.1 m and

lies on x-y plane. There is a magnetic field B = (50T) k̂ . The ring rotates with an angular velocity
 = 20 rad/sec about its axis. An external resistance of 10 is connected across the centre of the
ring and rim. Find the current through external resistance.
40. In the figure there are two identical conducting rods each of length ‘a’ rotating with angular
speed  in the directions shown. One end of each rod touches a conducting ring. Magnetic
field B exists perpendicular to the plane of the rings. The rods, the conducting rings and the
lead wires are resistanceless. Find the magnitude and direction of current in the resistance R.

B B

D
C R
41. A wire shaped as a semi-circle of radius a rotates about an axis OO' with an angular velocity
 in a uniform magnetic field of induction B (figure). The rotation axis is perpendicular to the
field direction. The total resistance of the circuit is equal to R. Neglecting the magnetic field of
the induced current, find the mean amount of thermal power being generated in the loop
during a rotation period.

a 

0 +B 0'
R

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EMI P. No.- 8
42. A conducting disc of radius R is rolling without sliding on a horizontal surface with a constant
velocity ' v ' . A uniform magnetic field of strength B is applied normal to the plane of the disc.
Find the EMF induced between ( C is centre, P&Q are opposite points on vertical diameter
of the disc)

(a) P& Q (b) P & C. (c) Q & C

43. A horizontal wire is free to slide on the vertical rails of a conducting frame as shown in figure. The
wire has a mass m and length l and the resistance of the circuit is R. If a uniform magnetic field B
is directed perpendicular to the frame,then find the terminal speed of the wire as it falls under the force
of gravity.

44. A pair of parallel horizontal conducting rails of negligible resistance shorted at one end is fixed on
a table. The distance between the rails is L. A conducting massless rod of resistance R can slide on
the rails frictionlessly. The rod is tied to a massless string which passes over a pulley fixed to the
edge of the table. A mass m, tied to the other end of the string hangs vertically. A constant magnetic
field B exists perpendicular to the table. If the system is released from rest, calculate:

(i) the terminal velocity achieved by the rod.

(ii) the acceleration of the mass at the instant when the velocity of the rod is half the terminal velocity.
45. A magnetic field B = (B0y / a) k is into the plane of paper in the +z direction. B0 and a are positive
constants. A square loop EFGH of side a, mass m and resistance R, in x-y plane, starts falling
under the influence of gravity. Note the directions of x and y axes in the figure. Find

(a) the induced current in the loop and indicate its direction,
(b) the total Lorentz force acting on the loop and indicate its direction,
(c) an expression for the speed of the loop, v(t) and its terminal value.

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EMI P. No.- 9
46. Two parallel vertical metallic rails AB & CD are separated by 1 m. They are connected at the two
ends by resistance R1 & R2 as shown in the figure. A horizontally metallic bar L of mass 0.2 kg
slides without friction, vertically down the rails under the action of gravity. There is a uniform
horizontal magnetic field of 0.6T perpendicular to the plane of the rails, it is observed that when the
terminal velocity is attained, the power dissipated in R1 & R2 are 0.76 W & 1.2 W respectively.
Find the terminal velocity of bar L & value R1 & R2.

47. A wire of mass m and length  can slide freely on a pair of smooth, vertical rails (figure). A magnetic
field B exists in the region in the direction perpendicular to the plane of the rails. The rails are
connected at the top end by an initially uncharged capacitor of capacitance C. Find the velocity of
the wire neglecting any electric resistance. (initial velocity of wire is zero)
× × C× ×

× × × ×
× × × ×

× × × ×

× × × ×

× × × ×
48. In the figure shown a conducting rod of length  , resistance R & mass m is moved with a
constant velocity v . The magnetic field B varies with time t as B = 5 t . At t = 0 the area of
the loop containing capacitor and the rod is zero and the capacitor is uncharged . The rod
started moving at t = 0 on the fixed smooth conducting rails which have negligible resistance.
Find :
..........
v=constant
(v=fu; r )
C B l
m
..........

(i) The current in the circuit as a function of time t .

(ii) If the above system is kept in vertical plane such that the rod can move vertically downward
due to gravity and other parts are kept fixed and B = constant = B0 , then find the maximum
current in the circuit .
49. Two parallel long smooth conducting rails separated by a distance  are connected by a
movable conducting connector of mass 'm'. Terminals of the rails are connected by the resistor
R & the capacitor C as shown. A uniform magnetic field B perpendicular to the plane of the
rails is switched on. The connector is dragged by a constant force F. Find the speed of the
connector as function of time if the force F is applied at t = 0. Also find the terminal velocity
of the connector.
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EMI P. No.- 10

F
R  B B C

50. A metal rod OA of mass m & length r is kept rotating with a constant angular speed  in a vertical
plane about a horizontal axis at the end O. The free end A is arranged to slide without friction along
a fixed conducting circular ring in the same plane as that of rotation. A uniform & constant magnetic

induction B is applied perpendicular & into the plane of rotation as shown in figure. An inductor L
and an external resistance R are connected through a switch S between the point O & a point C on
the ring to form an electrical circuit. Neglect the resistance of the ring and the rod. Initially, the
switch is open.

(a) What is the induced emf across the terminals of the switch ?
(b) (i) Obtain an expression for the current as a function of time after switch S is closed.
(ii) Obtain the time dependence of the torque required to maintain the constant angular speed,
given that the rod OA was along the positive X-axis at t = 0.
52. A circular loop of radius 1m is placed in a varying magnetic field given as B = 6t Tesla.
(a) Find the emf induced in the coil if the plane of the coil is perpendicular to the magnetic field.
(b) Find the electric field in the tangential direction, induced due to the changing magnetic field.
(c) Find the current in the loop if its resistance is1/m.
53. The current in an ideal, long solenoid is varied at a uniform rate of 0.01 A/s. The solenoid has 2000
turns/m and its radius is 6.0 cm. (a) Consider a circle of radius 1.0 cm inside the solenoid with its
axis coinciding with the axis of the solenoid. Write the change in the magnetic flux through this circle
in 2.0 seconds. (b) Find the electric field induced at a point on the circumference of the circle. (c)
Find the electric field induced at a point outside the solenoid at a distance 8.0 cm from its axis.
54. There exists a uniform cylindrically symmetric magnetic field directed along the axis of a cylinder
but varying with time as B = kt. If an electron is released from rest in this field at a distance of ‘r’
from the axis of cylinder, its acceleration, just after it is released would be (e and m are the electronic
charge and mass respectively)
55. A variable magnetic field creates a constant emf E in a conductor ABCDA. The resistances of
portion ABC, CDA and AMC are R1, R2 and R3 respectively. What current will be shown by
meter M? The magnetic field is concentrated near the axis of the circular conductor.

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EMI P. No.- 11
56. A charged ring of mass m = 50 gm, charge 2 coulomb and radius R = 2m is placed on a smooth
horizontal surface. A magnetic field varying with time at a rate of (0.2 t) Tesla/sec is applied on to the
ring in a direction normal to the surface of ring. Find the angular speed attained in a time t1 = 10 sec.
57. A triangular wire frame (each side = 2m) is placed in a region of time variant magnetic field having
dB/dt = 3 T/s. The magnetic field is perpendicular to the plane of the triangle. The base of the
triangle AB has a resistance 1  while the other two sides have resistance 2 each. The magnitude
of potential difference between the points A and B will be .......


58. A uniform magnetic field B fills a cylindrical volumes of radius R. A metal rod CD of length l is
placed inside the cylinder along a chord of the circular cross-section as shown in the figure. If the
magnitude of magnetic field increases in the direction of field at a constant rate dB/dt, find the
magnitude and direction of the EMF induced in the rod.

L
59. Find the dimension of the quantity , where symbols have usual meaining.
RCV
60. Find the self inductance of a solenoid which has 10 turns per cm. Its length is 1m and radius 1
cm.
61. The network shown in Fig. is a part of a complete circuit. What is the potential difference VB
– VA, when the current  is 5A and is decreasing at a rate of 103 (A/s)?

62. Figure shows a part of a circuit. Find the rate of change of the current, shown.
5V 3V
2
10V 2A 10H 20V

63. In the circuit shown find (a) the power drawn from the cell, (b) the power consumed by the resistor
which is converted into heat and (c) the power given to the inductor.
1A
2H
5V 3

64. Find the energy stored in the magnetic field inside a volume of 1.00 mm3 at a distance of 10.0 cm
from a long wire carrying a current of 4 A.
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65. What is the magnetic energy density (in terms of standard constant & r) at the centre of a
circulating electron in the hydrogen atom in first orbit. (Radius of the orbit is r)
66. A solenoid has an inductance of 10 Henry and a resistance of 2 . It is connected to a 10 volt
battery. How long will it take for the magnetic energy to reach 1/4th of its maximum value?
67. Two inductances L1 & L2 are connected in series & are separated by a large distance.
(a) Show that their equivalent inductance is L1 + L2.
(b) Why must their separation be large?
68. In the given current, find the ratio of i1 to i2 where i1 is the initial (at t = 0) current and i2 is steady
state (at t = ) current through the battery.

69. In the circuit shown, initially the switch is in position 1 for a long time. Then the switch is shifted to
position 2 for a long time. Find the total heat produced in R2.

70. Two resistors of 10 and 20 and an ideal inductor of 10H are connected to a 2V battery as
shown. The key K is shorted at time t = 0. Find the initial (t = 0) and final (t ) currents through
battery.

71. In the circuit shown in the figure the switched S1 and S2 are closed at time t = 0. After time t = (0.1)
ln 2 sec, switch S2 is opened. Find the current in the circuit at time t = (0.2) ln 2 sec.

72. Find the values of i1 and i2

(i) immediately after the switch S is closed.
(ii) long time later, with S closed.
(iii) immediately after S is open.
(iv) long time after S is opened.
73. Suppose the emf of the battery, the circuit shown varies with time t so the current is given by i(t) =
3 + 5t, where i is in amperes & t is in seconds. Take R = 4, L = 6H & find an expression for the
battery emf as function of time.

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EMI P. No.- 13
74. An emf of 15 volt is applied in a circuit containing 5 H inductance and 10  resistance. Find the
ratio of the currents at time t =  and t = 1 second.
75. In the circuit shown in figure switch S is closed at time t = 0. Find the
charge which passes through the battery in one time constant.
76. In a L–R decay circuit, the initial current at t = 0 is I. Find the total charge that has flown through the
resistor till the energy in the inductor has reduced to one–fourth its initial value.
77. In the LR circuit shown, what is the variation of the current I as a function of time? The switch is
closed at time t = 0 sec.

78. An inductor of inductance L = 400 mH and resistors of resistances R1 = 2 and R2 = 2 are

connected to a battery of e.m.f. E = 12V as shown in the figure. The internal resistance of the
battery is negligible. The switch S is closed at time t = 0. What is the potential drop across L as a
function of time? After the steady state is reached, the switch is opened. What is the direction and
the magnitude of current through R1 as a function of time?

79. A closed circuit consists of a source of constant emf E and a choke coil of inductance L
connected in series. The active resistance of the whole circuit is equal to R. It is in steady
state. At the moment t = 0 the choke coil inductance was decreased abruptly  times. Find the
current in the circuit as a function of time t.
80. A conducting frame ABCD is kept in a vertical plane. A conducting rod EF of mass m can
slide smoothly on it remaining horizontal always. The resistance of the loop is negligible and
inductance is constant having value L. The rod is left from rest and allowed to fall under
gravity and inductor has no initial current. A uniform magnetic field of magnitude B is present
throughout the loop pointing inwards. Determine.

(a) position of the rod as a function of time assuming initial position of the rod to be
x = 0 and vertically downward as the positive X-axis.
(b) maximum current in the circuit
(c) maximum velocity of the rod.
81. The average emf induced in the secondary coil is 0.1 V when the current in the primary coil
changes from 1 to 2 A in 0.1 s . What is the mutual inductance of the coils.
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82. Two coils, 1 & 2, have a mutual inductance = M and resistances R each. A current flows in coil 1,
which varies with time as: I1 = kt2 , where K is a constant and 't' is time. Find the total charge that
has flown through coil 2, between t = 0 and t = T.
83. A square loop of side 'a' with a capacitor of capacitance C is located between two current carrying
long parallel wires as shown. The value of I in the is given as I = I0sint.

(a) calculate maximum current in the square loop.

(b) Draw a graph between charge on the lower plate of the capacitor v/s time.
84. A long solenoid of radius a and number of turns per unit length n is enclosed by cylindrical shell of
radius R, thickness d (d <<R) and length L. A variable current i = i0sin t flows through the coil.
If the resistivity of the material of cylindrical shell is , find the induced current in the shell.

85. A very small circular loop of radius a is initially coplanar & concentric with a much larger circular
loop of radius b (> > a). A constant current I is passed in the large loop which is kept fixed in space
& the small loop is rotated with constant angular velocity  about a diameter. The resistance of the
small loop is R & its inductance is negligible.
(a) Find the current in the small loop as a function of time.
(b) Calculate how much torque must be exerted on the small loop to rotate it
(c) Calculate the induced emf in the large loop due to current(found in part (a)) in smaller loop
as a function of time.
86. In the figure shown two loops ABCD & EFGH are in the same plane . The smaller loop
carries time varying current  = b t, where b is a positive constant and t is time . The resistance
of the smaller loop is r and that of the larger loop is R. Find :

(i) induced current in the larger loop

(ii) the magnetic force on the loop EFGH due to loop ABCD .
87. A capacitor C with a charge Q0 is connected across an inductor through a switch S. If at t = 0, the
switch is closed, then find the instantaneous charge q on the upper plate of capacitor.

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EMI P. No.- 15
88. An inductor of inductance 2.0mH,is connected across a charged capacitor of capacitance 5.0F,and
the resulting LC circuit is set oscillating at its natural frequency. Let Q denote the instantaneous
charge on the capacitor, and I the current in the circuit .It is found that the maximum value of Q is
200 C.
(a) when Q=100C,what is the value of dI / dt ?
(b) when Q=200 C ,what is the value of I ?
(c) Find the maximum value of I.
(d) when I is equal to one half its maximum value, what is the value of Q
89. In the circuit shown switches S1 and S3 have been closed for 1 sec and S2 remained open. Just
after 1 second is over switch S2 is closed and S1, S3 are opened. Find after that instant

(a) the maximum current in the circuit containing inductor and capacitor only
(b) the maximum charge on the capacitor
(c) the charge on the upper plate of the capacitor as function of time taking the instant of
switching on of S2 and switching off all the switches to be t = 0.

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EMI P. No.- 16

ANSWER KEY
1. (i) 1.2 Volt (ii) 1.4 volt (iii) 17.5 C (iv) 3.5 A (v) 86/3 joule.
2. (a) In the round conductor the current flows clockwise, there is no current in the connector;
(b) in the outside conductor clockwise;
(c) in both round conductors, clockwise; no current in the connector,
(d) in the left-hand side of the figure eight, clockwise.

(a) (b) (c) (d)

3. Opposite direction, Same direction. 4. 2.5 mV 5. zero
6. 493 V 7. 1.6×10–5A 8. 3V, clockwise

9. 5.0 V 10. , 2J

 0ia 2 
11. 12. (a) 2.0 × 10–3 V (b) zero (c) 50 C (d)× 10–3 sin (10t)
2Rb

dB 2
13. i = (3/2)  t = 12 mV.. 14. C a2 R
dt

4 1  0 hi m N b
15. 2 × 10-5 × loge volts, clockwise 16. eim = p a2 N w B0 17. ln
3 3 2 a

 0 ia µ i a  a  b   2t   5µ20 i02 a 2    a  b  2

ab   n
18. (a) 
2
n ; (b)  = 0 0 n b  sin T  (c) heat =  Tr    b 
 b  T        

19. i = 1/2 (–1) n Ba2 t, where n = 1, 2, .... is the number of the half-revolution that the loop

performs at the given moment t. The plot i (t) is shown in figure where tn = 2n /  .
i

0 t
t1 t2 t3 t4

t4 < t3 < t2 < t1

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EMI P. No.- 17

 0 i 0 W  cos  t  L2  7 3 1
20. n  1 21. IEA= A ; IBE= A ; IFE = A
4  2  22 11 22
Y 

22. (i) 2.4 × 105 V (ii) from c to b

23. (a) zero (b) vB (bc), positive at b (c) vB(bc), positive at a (d) zero
B 0 v 0L
24. 3  10 –2 V 25. 26. (a) 4 m/s (b) 4 V (c) 3 V (d) 1 V.
2

27. (BV sin ) / r(1 + sin )

28. (a) at the ends of the diameter perpendicular to the velocity, 2rvB
(b) at the ends of the diameter parallel to the velocity, zero.
29. VyB0 30. By 8a / k 31. 2N 32. 10 V

33. 34. (i) 85.22 Tm2; (ii) 56.8 V; (iii) linearly

35. (i) 3.3 × 108 A, (ii) 1.1 × 1017 W, (iii) totally unrealistic
2
0 0 v b v   00 b
36. (a)  = n (b) F =  n  (c) (b – a) / loge (b /a) from the long wire.
2R a R  2 a

 1
37. 67.5 mV 38. B g sin 39. A
2 3

Ba 2 BRv 3 BRv

40. from C to D 41. <P> = (a2B)2/8R. 42. (a) 2BR v (b) (c)
R 2 2

mgR mgR g
43. 44. (i) Vterminal = ; (ii)
B 2l 2 B2 Z2 2
B0av
45. (a) i = in anticlockwise direction, v = velocity at time t,
R
 B 2a 2 t 
mgR   0 
(b) Fnett=B02a2V/R, (c) V = 2 2 1  e mR 
B0 a  
 
mgt
46. V = 1 ms1, R1 = 0.47 , R2 = 0.30  47.
m  CB2  2

mg B  c
48. (i) i = 10  v c (1  e t/Rc) (ii) imax =
m  B22c
  B 2  2 t 
FR   2 2 
 R (m  B  C )  FR
 1  e 
49. v= B2 2  , vterminal =
B22
 

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EMI P. No.- 18

50. (a) E =
1
Br2 (b) (i) I =

Br 2 1e  Rt / L
, (ii)  =

mgr
cos t +
B2 r 4
(1  eRt/L)
2 2R 2 4R
52. (a) 6 Volt (b) 3 N/C (c) 3 A
2 –10
53. (a) 16 × 10 = 1.6 × 10 Weber (b) 4 × 10–8 V/m (c) 18 × 10–8 = 5.6 × 10–7 V/m
–8

erk
54. directed along tangent to the circle of radius r, whose centre lies on the axis of cylinder..
2m
E R1
55. R1R 2  R 2 R 3  R 3R 1 56. 200 rad/sec 57. 0.4 V

2
l dB 2 l
58. R  59. I–1 60. 4×10-4 H
2 dt 4
61. 15V 62. 2.2 A/s, decreasing 63. (a) 5 W (b) 3W (c) 2 W
0 e 4
64. 2.55 × 10 –14
J 65. 66. t = (L/R) n 2 = 3.47 s
128 3  0mR 5
67. (b) Separation is large to neglect mutual inductance 68. 0.8
LE 2 1 1
69. 70. A, A 71. 67/32 A
2R 12 15 10
72. (i) i1 = i2 = 10/3 A, (ii) i1 = 50/11 A ; i2 = 30/11 A, (iii) i1 = 0, i2 = 20/11 A, (iv) i1 = i2 = 0

e2 EL
73. 42 + 20t volt 74. 75.
e2 1 eR 2
Rt
V L
76. L I 2R 77. – e 78. 12e–5t, 6e–10t
R
E g
[1  cos t ] , (b) I
2mg g
79. = [1 + ( – 1) e–tR/L 80. (a) x = 2 = , (c) Vmax =
R  max
B 
2
81. 0.01 H 82. kMT /(R)

0a ( 0 ni 0  cos t )a 2 (Ld)

83. (a) Imax = CI 02ln 2 , (b) 84. I=
  2R

2 2
a 2  0 I sin t   a 2  0 I sin  t   a 2  0   I cos 2 t
85. (a) i = (b) = R  2 b  (c) 
 2b 

R
2bR    

0 a b 4  20 ab 4  1 
86. i = n ; F = 12 2 R n 87. q = Q0sin  LC t  2 
2 R 3 3  
88. (a)104A/s (b) 0 (c) 2A (d) 100 3 C
89. (a) (1-1/e) = 0.89 amp, (b) (1-1/e) = 0.89 Coul.

 
(c) 2 (1–e–1) cos  t  4  = 0.89 cos  t   
   4
JRS Tutorials, Durgakund, Varanasi-221005, Ph No.(0542) 2311922, 9794757100